In mathematics, a Lie algebra is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. The Lie bracket of two vectors and is denoted . The vector space together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative.
2D computer graphics is the computer-based generation of digital images—mostly from two-dimensional models and by techniques specific to them. It may refer to the branch of computer science that comprises such techniques or to the models themselves.
In Euclidean geometry, an affine transformation or affinity is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize a multivariate quadratic function subject to linear constraints on the variables. Quadratic programming is a type of nonlinear programming.
In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces.
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.
In linear algebra, the column space of a matrix A is the span of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation.
In mathematics, the cross product or vector product is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space, and is denoted by the symbol . Given two linearly independent vectors a and b, the cross product, a × b, is a vector that is perpendicular to both a and b, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product.
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of orthogonal matrices, where the group operation is given by matrix multiplication. The orthogonal group is an algebraic group and a Lie group. It is compact.
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.
In mathematics, the adjoint representation of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is , the Lie group of real n-by-n invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible n-by-n matrix to an endomorphism of the vector space of all linear transformations of defined by: .
A 3D projection is a design technique used to display a three-dimensional (3D) object on a two-dimensional (2D) surface. These projections rely on visual perspective and aspect analysis to project a complex object for viewing capability on a simpler plane.
In linear algebra and functional analysis, a projection is a linear transformation from a vector space to itself such that . That is, whenever is applied twice to any vector, it gives the same result as if it were applied once. It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.
In affine geometry, uniform scaling is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions. The result of uniform scaling is similar to the original. A scale factor of 1 is normally allowed, so that congruent shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a photograph, or when creating a scale model of a building, car, airplane, etc.
In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map L : V → W between two vector spaces V and W, the kernel of L is the vector space of all elements v of V such that L(v) = 0, where 0 denotes the zero vector in W, or more symbolically:
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by , is the factor by which the eigenvector is scaled.
In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of 1/2. The spin number describes how many symmetrical facets a particle has in one full rotation; a spin of 1/2 means that the particle must be rotated by two full turns before it has the same configuration as when it started.
The direct-quadrature-zerotransformation or zero-direct-quadraturetransformation is a tensor that rotates the reference frame of a three-element vector or a three-by-three element matrix in an effort to simplify analysis. The DQZ transform is the product of the Clarke transform and the Park transform, first proposed in 1929 by Robert H. Park.
